# Fractions - Division

In this article, we will learn dividing fractions according to the syllabus of Primary 6 / P6 Maths grade. In the division of fractions, we will cover the following subtopics.

- Dividing a proper fraction by a whole number
- Dividing a whole number by a proper fraction
- Dividing a proper fraction by a proper fraction
- Word problems involving the division of fractions

We will also be going through a quick revision of the multiplication of fractions.

**Multiplication Of Fractions**

In this, we will go through the two methods used in the multiplication of fractions.

**Practice Problems**

**Question 1: **

Find the value of \(\begin{align*} \frac{7}{10} \times \frac {4}{21} \end{align*}\) in its simplest form.

**Solution:**

**Method 1: **

We multiply the numerator by the numerator to get a new numerator and then the denominator by the denominator to get a new denominator.

Always remember to express the fraction in your answer in its simplest form.

\(\begin{align*} \frac{7} {10} \times \frac {4}{21} &= \frac{7 \times 4} {10 \times 21} \\ &=\frac{28}{ 210}\\ &=\frac{4}{ 30}\\ &=\frac{2}{ 15} \end{align*}\)

**Method 2: **

We cross-divide and simplify the fractions to their simplest form before we multiply the numerators and then the denominators.

\(\begin{align*} \frac{7} {10} \times \frac {4}{21} &=\frac{1} {5} \times \frac {2}{3} \\ &=\frac{2}{ 15} \end{align*}\)

**Answer:**

\(\begin{align} \frac{2} {15} \end{align} \)

**Question 2: **

Find the value of \(\begin{align*} \frac {8}{21} \times 14 \end{align*}\). Express your answer in its simplest form.

**Solution: **

\(\begin{align*} \frac {8}{21} \times 14 &= \frac {8}{21} \times \frac {14}{1} \\ &= \frac {8\times 14}{21} \\ &= \frac {8\times 2}{3} \\ &= \frac {16}{3} \\ &= 5\frac {1}{3} \end{align*}\)

**Answer:**

\(\begin{align*} 5\frac {1}{3} \end{align*}\)

**Question 3: **

Find the value of \(\begin{align*} \frac {16}{3} \times \frac {9}{10} \end{align*}\) in its simplest form.

**Solution: **

\(\begin{align*} \frac {16}{3} \times \frac {9}{10} &= \frac {16 \times 9}{3 \times 10} \\ &= \frac {8 \times 3}{1 \times 5} \\ &= \frac {24}{5} \\ &= 4\frac {4}{5} \\ \end{align*}\)

**Answer:**

\(\begin{align*} 4\frac {4}{5} \end{align*}\)

**1. Divide A Proper Fraction By A Whole Number**

In this division, we will take a proper fraction and divide it by a whole number. To express a whole number as a fraction, we express it as a fraction with the denominator ‘\(1\)’.

**Practice Problems**

**Question 1: **

\(\begin{align*} \frac{2} {5} \end{align*}\) of the cake was shared equally with \(2\) boys. What fraction of the cake did each boy get?

**Solution:**

Fraction of cake to be shared with \(2\) boys \(\begin{align*} =\frac{2} {5} \end{align*} \)

**K**eep the first fraction.

**C**hange \(\div\) to \(\times\)

**F**lip the second fraction

\(\begin{align*} \frac{2} {5} \div \frac{2} {1} &=\frac{2} {5} \times \frac{1} {2} \\ &=\frac{1} {5} \end{align*}\)

Each boy gets \(\begin{align*} \frac{1} {5} \end{align*}\) of the cake.

**Answer:**

\(\begin{align*} \frac{1} {5} \end{align*}\)

**Question 2: **

**\(\begin{align*} \frac{7} {12} \div 4 \end{align*}\)** is the same as __________

**Solution: **

\(\begin{align*} \frac{7} {12} \div \frac{4} {1} &= \frac{7} {12} \times \frac{1} {4} \\ &= \frac{7}{48} \end{align*}\)

**Answer:**

\(\begin{align*} \frac{7}{48} \end{align*}\)

**2. Divide Whole Number By A Proper Fraction**

In this concept, we take a whole number and divide it by a proper fraction.

**Practice Problems**

**Question 1: **

How many \(\begin{align*} \frac{4}{9} \;m \end{align*}\) tapes can be cut out from a tape that is \(12 \;m\) long?

**Solution:**

Total length of tape \(=12 \;m\)

Length of each tape \(\begin{align*} =\frac{4}{9} \;m \end{align*}\)

Using the **KCF method**,

\(\begin{align*} 12 \;m \div \frac{4}{9} \;m &= \frac{12}{1} \;m \times \frac{9}{4} \;m \\\\ &= 27 \;\text{pieces} \end{align*}\)

**Answer:**

\(\begin{align*} 27 \;\text{pieces} \end{align*}\)

**Question 2: **

\(\begin{align*} 8 \div \frac{7}{11} = \end{align*}\) __________

**Solution: **

\(\begin{align*} 8 \div \frac{7}{11} &= \frac{8}{1} \times \frac {11}{7} \\ &=\frac{88}{7}\\ &= 12\frac{4}{7} \end{align*}\)

**Answer:**

\(\begin{align*} 12\frac{4}{7} \end{align*}\)

**3. Divide Proper Fraction By Proper Fraction**

**Practice Problems**

**Question 1: **

Jamson bought a bag of sugar with a mass of \(\begin{align*} \frac{9}{10} \;kg \end{align*} \). He repackaged the sugar into smaller packs, each containing \(\begin{align*} \frac{3}{20} \;kg \end{align*} \) of sugar. How many packs of sugar were there?

**Solution: **

Total mass of sugar \(\begin{align*} =\frac{9}{10} \;kg \end{align*} \)

Mass of \(1\) small pack of sugar \(\begin{align*} =\frac{3}{20} \;kg \end{align*} \)

Using the **KCF method**,

\(\begin{align*} \frac{9}{10} \div \frac{3}{20} &= \frac{9}{10} \times \frac{20}{3}\\\\ &= 6 \;\text{packs of sugar} \end{align*}\)

**Answer:**

\(6\) packs of sugar

**Question 2: **

Evaluate \(\begin{align*} \frac{7}{13} \div \frac{5}{6} \end{align*} \)

**Solution:**

\(\begin{align*} \frac{7}{13} \div \frac{5}{6}&= \frac{7}{13} \times \frac{6}{5}\\ &= \frac{42}{65} \end{align*}\)

**Answer:**

\(\begin{align*} \frac{42}{65} \end{align*}\)

**4. Word Problems - Four Operations**

We will be solving the word problems on fractions involving the four operations i.e addition, subtraction, multiplication, and division.

**Practice Problems**

**Question 1: **

A piece of cloth \(\begin{align*} \frac{3}{4} \;m \end{align*}\) long is cut into \(12\) equal pieces of the same length.

Find the length of one small piece of cloth.

**Solution: **

Total length of the cloth \(\begin{align*} =\frac{3}{4} \;m \end{align*}\)

Number of equal pieces \(= 12\)

Length of one piece \(\begin{align*} =\frac{3}{4} \;m \div 12 \end{align*}\)

\(\begin{align*} &=\frac{3}{4} \;m \times \frac{1}{2} \\ &= \frac{1}{16} \;m \end{align*}\)

**Answer:**

\(\begin{align*} \frac{1}{16} \;m \end{align*}\)

**Question 2: **

\(15\) cookies were distributed equally in class. Each student received \(\begin{align*} \frac{5} {7} \end{align*}\) of the cookie. How many students were there in the class?

**Solution: **

Total cookies distributed \(=15\) cookies

Fraction of cookies each student received \(\begin{align*} =\frac{5} {7} \end{align*}\)

Total number of students \(\begin{align*} =15 \div \frac{5} {7} \end{align*}\)

\(\begin{align*} &=\frac{15}{1} \times \frac{7}{5}\\ &=21 \end{align*}\)

**Answer:**

\(21\) students

**Question 3: **

A tin contains \(\begin{align*} \frac{9}{10} \;kg \end{align*}\) of tea leaves. Mindy packed them into smaller packets each containing \(\begin{align*} \frac{2}{15} \;kg \end{align*}\) of tea leaves. What would be the maximum number of packets that Mindy could get?

**Solution: **

Total mass of tea leaves \(\begin{align*} =\frac{9}{10} \;kg \end{align*}\)

Mass of tea leaves in \(1\) packet \(\begin{align*} = \frac{2}{15} \;kg \end{align*}\)

Total number of packets \(\begin{align*} =\frac{9}{10} \div \frac{2}{15} \end{align*}\)

\(\begin{align*} &=\frac{9}{10} \times \frac{15}{2} \\ &=\frac{27}{4} \\ &= 6\frac{3}{4} \end{align*}\)

Thus, the maximum number of packets that Mindy can get is \(6\) packets.

**Answer:**

\(6\) packets

**Question 4: **

What is the fraction represented by \(A\) in the figure below?

**Solution: **

Strategy is to count the number of gaps.

Value of \(4\) gaps \(\begin{align*} = 1\frac{1}{2}-1\frac{1}{4} \end{align*} \)

\(\begin{align*} &= \frac{3}{2}-\frac{5}{4} \\ &= \frac{1}{4} \end{align*}\)

Value of \(1\) gap \(\begin{align*} = \frac{1}{4} \div 4 \end{align*}\)

\(\begin{align*} = \frac{1}{16} \end{align*}\)

Therefore,

\(\begin{align*} A &=1\frac{1}{2}+ \frac{1}{16 } \\ &= \frac{25}{16 } \\ &=1\frac{9}{16 } \end{align*}\)

**Answer: **

\(\begin{align*} 1\frac{9}{16 } \end{align*}\)

**Question 5: **

Timothy bought \(\begin{align*} \frac{1}{4} \;kg \end{align*}\) of grapes. He ate \(\begin{align*} \frac{1}{6} \end{align*} \) of them and packed the remaining grapes into \(5\) bags. If each bag of grapes had the same mass, what was the mass of each bag of grapes?

**Solution: **

Total mass of grapes \(\begin{align*} =\frac{1}{4} \;kg \end{align*}\)

Mass of grapes eaten \(\begin{align*} =\frac{1} {6} \times \frac{1} {4} \;kg \end{align*}\)

\(\begin{align*} =\frac{1}{24} \;kg \end{align*} \)

Mass of the remaining grapes \(\begin{align*} =\frac{5} {6} \times \frac{1} {4} \;kg \end{align*}\)

\(\begin{align*} =\frac{5} {24} \;kg \end{align*}\)

Total number of bags \(= 5\)

Mass of each bag of grapes \(\begin{align*} =\frac{5} {24} \;kg \div 5 \end{align*} \)

\(\begin{align*} &= \frac{5} {24} \;kg \times \frac{1}{5} \\ &= \frac{1} {24} \;kg \end{align*}\)

**Answer:**

\(\begin{align*} \frac{1} {24} \;kg \end{align*}\)

**Conclusion**

In this article, we have recapped the concept of multiplication of fractions and learned the concept of division of fractions.

For doing the division of fractions, we have learned the following KCF strategy:

- Keep the first fraction
- Change the into
- Flip the second fraction

You should now be able to do the following:

- Division of fraction by a whole number
- Division of the whole number by a fraction
- Division of fraction by a fraction
- Word problems involving the division of fractions

**Test Yourself**

\(\frac{7}{12} \div 4\) is same as __________.

Ans: (1) 1 / 4 × 7 / 12

8 / 5 ÷ 15 / 8 = __________.

Ans: (3) 64 / 75

Which of the following is not equal to 2 / 6?

Ans: (3) 1 / 3 + 1 / 3